By Richard Evan Schwartz
A polytope alternate transformation is a (discontinuous) map from a polytope to itself that may be a translation anywhere it really is outlined. The 1-dimensional examples, period trade variations, were studied fruitfully for a few years and feature deep connections to different parts of arithmetic, resembling Teichmuller thought. This booklet introduces a basic technique for developing polytope alternate modifications in larger dimensions after which reviews the best instance of the development intimately. the best case is a 1-parameter kin of polygon alternate alterations that seems to be heavily concerning outer billiards on semi-regular octagons. The 1-parameter family members admits a whole renormalization scheme, and this constitution makes it possible for a pretty entire research either one of the procedure and of outer billiards on semi-regular octagons. the fabric during this e-book was once came across via machine experimentation. however, the proofs are conventional, apart from a number of rigorous computer-assisted calculations
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In fact, our calculation will show that all points in D3 and D5 eventually do return to these regions. This calculation is part of Calculation 11, described below. 50 5. OUTER BILLIARDS ON SEMIREGULAR OCTAGONS It is tempting to show that the ﬁrst return map to Dn is conjugate to the octagonal PET at the corresponding parameter. However, this does not work directly. We need to massage the picture a bit to make things work. 9) where Dn0 is the lower dogbone and Dn1 is the upper dogbone. There is an obvious involution ι : Dn → Dn which simply interchanges the two dogbones using a piecewise translation.
Restriction to Loops: Suppose we ﬁx a vertex basepoint x ∈ Γ and let X be the corresponding polytope. Let LOOP(Γ, x) be the semigroup of loops in Γ which start and end at x. The functor above restricts to a map LOOP(Γ, x) → PET(X). 27 28 3. MULTIGRAPH PETS Here PET(X) is interpreted as the semigroup of PETs with domain X. 3. Back to Earth: Here is a more concrete description of the whole construction. , 2n−1 , x2n = x, where k is an edge connecting xj−1 to xj+1 . , X2N = X be the corresponding polytopes.
The tiles involved in the continued fraction conjecture have been shaded. ” 33 34 4. 1: Tiling of the unit square for s = 36/61. We will not really study the dynamics of the alternating grid systems in this monograph, and in particular we will not make any progress towards proving the continued fraction conjecture – we don’t know how to prove it. Our main purpose here is to show the connection between the alternating grid system, double lattice PETs, and the octahedral PETs. However, our interest in proving the continued fraction conjecture lead to our discovery of the octagonal PETs.
The Octagonal PETs by Richard Evan Schwartz